Longtime behavior of nonlocal Cahn-Hilliard equations

By Ciprian G. Gal, Maurizio Grasselli. 2012

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Here we consider the nonlocal Cahn-Hilliard equation with constant mobility in a bounded domain. We prove that the associated dynamical system has an exponential attractor, provided that the potential is regular. In order to do that a crucial step is showing the eventual boundedness of the order parameter uniformly with respect to the initial datum. This is obtained through an Alikakos-Moser type argument. We establish a similar result for the viscous nonlocal Cahn-Hilliard equation with singular (e.g., logarithmic) potential. In this case the validity of the so-called separation property is crucial. We also discuss the convergence of a solution to a single stationary state. The separation property in the nonviscous case is known to hold when the mobility degenerates at the pure phases in a proper way and the potential is of logarithmic type. Thus, the existence of an exponential attractor can be proven in this case as well.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Materials Chemistry

View Paper: https://www.semanticscholar.org/paper/f7df00d245c9bb45de90810c14706120f6688d80

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In Applied Mathematics

In Computational Theory and Mathematics

In Materials Chemistry

Citations in the corpus

The nonlocal Cahn–Hilliard equation with singular potential: Well-posedness, regularity and strict separation property (2017, background): “We are now in a position to carry out an iterative argument (see [35] and references therein).”